*All truths are easy to understand once they are
discovered; the point is to discover them.*

attributed to Galileo Galilei

# Research interests

I am a geometric topologist working on knot theory and low-dimensional manifolds.

More generally, my mathematical interests include

- topology and algebraic topology,
- group theory and cohomology of groups,
- braid groups and representation theory,
- algorithms and computer algebra,
- game theory and social choice theory.

The following remarks outline my research activities, with links to my publications. My mémoire d'habilitation provides a synopsis of my research up to 2007.

*C'est par la logique que l'on prouve
et par l'intuition que l'on découvre.*

Henri Poincaré

## Finite type invariants of knots and links

For several years I have been very much interested in the Vassiliev theory of knots in 3-dimensional manifolds. (A good place to start reading about this topic is Dror Bar-Natan's homepage.) One of the central problems in this area is whether or not Vassiliev invariants distinguish all knots. Initially this question was raised by Vassiliev for knots in euclidean 3-space, for which it remains open to the present day.

In order to analyze this question in a broader context,
I proposed to consider it as a property of the ambient 3-manifold *M*.
The results that I obtained indicate that Vassiliev's question
is intimately related to the topology of *M*.

Firstly, rational Vassiliev invariants do not distinguish knots in S²×S¹. In the course of this investigation it turned out that Vassiliev invariants with values in a torsion group play a crucial role. This is the first appearance of torsion in the Vassiliev theory of knots. It is yet unknown whether this can occur for knots in euclidean 3-space.

Secondly, in every Whitehead manifold there exist knots that cannot be distinguished by Vassiliev invariants. It is natural to ask whether such a pathological situation can arise in a closed 3-manifold. This question has led to an unexpected connection with the Poincaré conjecture: if Vassiliev invariants distinguish knots in each homotopy sphere, then this implies the Poincaré conjecture (in other words every homotopy sphere is homeomorphic to the standard sphere). During this investigation I also developed a surgery proof of Bing's theorem characterizing the 3-sphere.

*Addendum.* To put things into perspective, let me repeat here what I have emphasized in my talks:
from its very beginning my approach was not intended as an attack on the Poincaré
conjecture, but rather to compare the relative difficulty of Vassiliev's question.
My result then says that Vassiliev's question (for closed 3-manifolds or even only for homotopy spheres)
is at least as difficult as the Poincaré conjecture. Unfortunately, my result does not establish
an equivalence. In the meantime the Poincaré conjecture has been proven
by Grigori Perelman,
building on work of Richard Hamilton.

*Anecdote.* My talk "Invariants de Vassiliev et Conjecture de Poincaré" on June 14, 2005,
happened to be the last Arnold
Seminar in Paris,
which I consider a great honour but sad nonetheless.
The theorem is reflected upon at the end of Arnold's article *
Forgotten and neglected theories of Poincaré*, Russ. Math. Surv. 61 (2006) 1-18.

## Surface invariants of finite type

Slice knots naturally appear in the study of singularities of surfaces in 4-space
(i.e. points where the surface is not locally flat).
Closely related is the stricter notion of ribbon knot, which spawned
Fox' notoriously difficult problem whether every smoothly slice knot is ribbon.
(See Charles Livingston's survey of classical knot concordance.)
These concepts have received much attention ever since the seminal work of Fox and Milnor
around 1960, who proved that the Alexander polynomial *Δ(K)* of a smoothly slice knot *K* has a beautiful
and very strong symmetry: *Δ(K) = f(t) f(1/t)* for some polynomial *f(t)*.
No similar result is yet known for the Jones polynomial.
My article on the Jones polynomial of ribbon links
can be seen as a first modest step in that direction.

This investigation led to the concept of surface invariants of finite type.
For example, if we expand the Jones polynomial *V(L)* into
a power series *Σ d _{k}(L) h^{k}* at

*t=-1*, instead of

*t=1*as usual, then the 0th coefficient is the determinant of the link

*L*obtained from a Seifert surface

*S*spanning it. The coefficients

*d*are not of finite type with respect to crossing changes of

_{k}(L)*L*, but surprisingly they turn out to be of finite type with respect to band crossing changes of the surface

*S*. This discovery promises to reconcile the classical theory of Seifert surfaces, or more generally immersed surfaces in

*R*, with the more recent theory of quantum invariants.

^{3}Motivated by the study of ribbon knots I became interested in symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka in 1957. It is fairly easy to see that every symmetric union represents a ribbon knot, but the converse question is still open. In joint work with Christoph Lamm, I have studied equivalence of symmetric union diagrams and constructed a refined Jones polynomial for symmetric unions.

## Knot group representations

I wrote my Ph.D. thesis under the supervision of Prof. Carl-Friedrich Bödigheimer at the University of Bonn about knot group representations and invariants of finite type. Part of this work grew out of a question of Daniel Altschuler and my answer to it: The number of knot group representations is not a Vassiliev invariant. Generalizing the technique of twist sequences employed in the proof has led to a a geometric characterization of Vassiliev invariants.

I went on to study algebraic invariants associated with finite knot group representations, the so-called knot colouring polynomials.

Knot quandles are an algebraic structure generalizing knot groups. I have achieved a complete calculation of the low-dimensional (co)homology of knot and link quandles, which entails in particular a homological characterization of the unknot. This could potentially be employed to tackle the unknotting problem: given a knot, how can we decide whether it is trivial or not? This problem is solvable in principle but we are still lacking an efficient algorithm (say polynomial in the number of crossings).

The quandle axioms model conjugation in a group, or equivalently the Reidemeister moves of knot diagrams. Even though this is a relatively weak algebraic structure, quandles lead to a surprisingly rich theory which has not ceased to swell in the last 25 years. (For the latest news see the arXiv.) They feature classifying spaces and a cohomology theory analogous to the case of groups (see Colin Rourke's homepage). My contribution is the theory of Quandle coverings and their Galois correspondence.

## Yang-Baxter deformations

Closely related to knot group representations and their colouring polynomials
is the deformation theory of certain Yang-Baxter operators.
I have studied such Yang-Baxter operators coming from a finite group *G*.
It has been long known that the associated knot invariant corresponds
to counting homomorphisms from the knot group into *G*.

More surprisingly, its deformations over a suitable ring
**Q**[*t*|*t ^{n}*=1] correspond
to knot colouring polynomials.
This point of view allows us to unify three a priori distinct families of invariants:
quandle state-sum invariants are a specialization of knot colouring polynomials,
which in turn are a special case of Yang-Baxter invariants.

On the other hand, I proved that an R-matrix of this type is usually rigid
in the sense that there are no deformations over the ring **Q**[[*t*]]
of formal power series. This is in sharp contrast to quantum invariants,
which arise from power series deformations of the trivial operator.
For details see my paper on Yang-Baxter deformations of quandles and racks.

I should emphasize that Yang-Baxter cohomology includes rack cohomology. The latter has been intensively studied in recent years and, in the modular case, leads to non-trivial and topologically interesting Yang-Baxter deformations. It follows that the Yang-Baxter cohomology of racks must be at least as complicated, and so the modular case stood out as a difficult yet promising problem. This has only recently been solved in the sequel Yang-Baxter deformations and rack cohomology.

## Algorithms and programming

Although I cannot be considered an applied mathematician, I am very much interested in algorithmic aspects of mathematics. This has led to a few publications oriented towards computer algebra.

While teaching programming and algorithmics to mathematics students at Grenoble University, I also became interested in enumeration techniques for diophantine equations, as explained in my article on bimonotone enumeration.

A more recent result is an effective version of the Fundamental Theorem of Algebra: my real-algebraic proof is based on Sturm chains, which makes it accessible even to the novice, and directly translates to a practical root-finding algorithm.

## Social Choice

I have also strolled into economics when venturing on the classification of arrovian juntas. (This is a remote echo of my game theory studies at the University of Bonn.)

If you are a topologist and you marvel at this bizarre digression, you may find some comfort in Beno Eckmann's pleasant note Social Choice and Topology, Expositiones Mathematicae 22 (2004), 385-393.

*An expert is a man who has made all the mistakes,
which can be made, in a very narrow field.*

Niels Bohr

*An expert is someone who knows some of the worst mistakes
that can be made in his subject, and how to avoid them.*

Werner Heisenberg